Finding Squares of numbers
To find the squares of numbers near numbers of which squares are known

To find 41^2 , Add 40+41 to 1600 =1681

To find 59^2 , Subtract 60^2-(60+59) =3481

Finding number of Positive Roots
If an equation (i:e f(x)=0 ) contains all positive co-efficient of any powers of x , it has no positive roots then.

Eg: x^4+3x^2+2x+6=0 has no positive roots .

Finding number of Imaginary Roots
For an equation f(x)=0 , the maximum number of positive roots it can have is the number of sign changes in f(x) ; and the maximum number of negative roots it can have is the number of sign changes in f(-x) .
Hence the remaining are the minimum number of imaginary roots of the equation(Since we also know that the index of the maximum power of x is the number of roots of an equation.)

Reciprocal Roots
The equation whose roots are the reciprocal of the roots of the equation ax^2+bx+c is cx^2+bx+a

Roots
Roots of x^2+x+1=0 are 1,w,w^2 where 1+w+w^2=0 and w^3=1Finding Sum of the rootsFor a cubic equation ax^3+bx^2+cx+d=o sum of the roots = - b/a sum of the product of the roots taken two at a time = c/a product of the roots = -d/a

For a biquadratic equation ax^4+bx^3+cx^2+dx+e = 0 sum of the roots = - b/a sum of the product of the roots taken three at a time = c/a sum of the product of the roots taken two at a time = -d/a product of the roots = e/a

Maximum/Minimum
-> If for two numbers x+y=k(=constant), then their PRODUCT is MAXIMUM if x=y(=k/2). The maximum product is then (k^2)/4

-> If for two numbers x*y=k(=constant), then their SUM is MINIMUM if x=y(=root(k)). The minimum sum is then 2*root(k) .

Problems on clocks
Problems on clocks can be tackled as assuming two runners going round a circle , one 12 times as fast as the other . That is , the minute hand describes 6 degrees /minute the hour hand describes 1/2 degrees /minute . Thus the minute hand describes 5(1/2) degrees more than the hour hand per minute .
The hour and the minute hand meet each other after every 65(5/11) minutes after being together at midnight. (This can be derived from the above) .

Co-ordinates
Given the coordinates (a,b) (c,d) (e,f) (g,h) of a parallelogram , the coordinates of the meeting point of the diagonals can be found out by solving for [(a+e)/2,(b+f)/2] =[ (c+g)/2 , (d+h)/2]

Ratio
If a1/b1 = a2/b2 = a3/b3 = .............. , then each ratio is equal to (k1*a1+ k2*a2+k3*a3+..............) / (k1*b1+ k2*b2+k3*b3+..............) , which is also equal to (a1+a2+a3+............./b1+b2+b3+..........)

-> The sum of an infinite GP = a/(1-r) , where a and r are resp. the first term and common ratio of the GP .

Mixtures
If Q be the volume of a vessel q qty of a mixture of water and wine be removed each time from a mixture n be the number of times this operation be done and A be the final qty of wine in the mixture then ,
A/Q = (1-q/Q)^n

Successive Discounts
Formula for successive discounts
a+b+(ab/100)
This is used for succesive discounts types of sums.like 1999 population increses by 10% and then in 2000 by 5% so the population in 2000 now is 10+5+(50/100)=+15.5% more that was in 1999 and if there is a decrease then it will be preceeded by a -ve sign and likewise.